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Research Papers

A Fluctuating Elastic Plate Model Applied to Graphene

[+] Author and Article Information
Xiaojun Liang

Department of Mechanical Engineering
and Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104

Prashant K. Purohit

Department of Mechanical Engineering
and Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104
e-mail: purohit@seas.upenn.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 24, 2016; final manuscript received May 18, 2016; published online June 9, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(8), 081008 (Jun 09, 2016) (8 pages) Paper No: JAM-16-1155; doi: 10.1115/1.4033681 History: Received March 24, 2016; Revised May 18, 2016

Over the past few decades, the measurement and analysis of thermal undulations has provided a route to estimate the mechanical properties of membranes. Theoretically, fluctuating elastic membranes have been studied mostly by Fourier analysis coupled with perturbation theory (to capture anharmonic effects), or by computer simulations of triangulated surfaces. These techniques as well as molecular dynamic simulations have also been used to study the thermal fluctuations of graphene. Here, we present a semi-analytic approach in which we view graphene as a triangulated membrane, but compute the statistical mechanical quantities using Gaussian integrals. The nonlinear coupling of in-plane strains with out-of-plane deflections is captured using a penalty energy. We recover well-known results for the scaling of the fluctuations with membrane size, but we show that the fluctuation profile strongly depends on boundary conditions and type of loading applied on the membrane. Our method quantitatively predicts the dependence of the thermal expansion coefficient of graphene on temperature and shows that it agrees with several experiments. We also make falsifiable predictions for the dependence of thermal expansion coefficient and the heat capacity of graphene on applied loads and temperature.

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References

Figures

Grahic Jump Location
Fig. 1

(a) Equilateral triangle discretization scheme of a square plate as in Ref. [17]. (b) For computing the energy cost of violating the compatibility constraint equation (3), we show the contour around a node whose out-of-plane displacement is w. Out-of-plane displacements of surrounding nodes are denoted by w1, w2, w3, w4, w5, and w6. Two other integration contours around neighboring nodes are shown.

Grahic Jump Location
Fig. 2

The upper surfaces show value of integral of compatibility equation around every node, with (a) one edge hinged and others free and (b) two edges hinged others free, respectively, for a fluid membrane with zero penalty energy. The lower surfaces (on the order of 10−6) show the same quantity for solid membranes with penalty energy scalar λ = 109 pN nm.

Grahic Jump Location
Fig. 3

((a) and (b)) Variance of fluctuating elastic plate with applied hydrostatic tension F, with one edge hinged and others free and two edges hinged others free, respectively. The upper surfaces in (a) and (b) show the fluctuation of a fluid membrane with the same mechanical properties and boundary conditions as our solid plates. (c) The solid lines associated with left axis show how the variance of the fluctuation becomes independent of the penalty energy parameter λ which enforces the compatibility constraint, and the dashed lines associated with right axis show how the average value of compatibility function fc over all nodes approaches zero as λ becomes sufficiently large. (d) w¯ of a fluctuating elastic plate as function of size with two different boundary conditions shown in (a) and (b). The parameters of the membrane are reported in groups C–I in Table 1. The inset corresponds to group I in Table 1.

Grahic Jump Location
Fig. 4

Distinct variance profiles of a sheet under shear loading as given in groups J–L of Table 1. (a) Mode n = 1, m = 1 has the largest amplitude. (b) Mode n = 1, m = 2 has the largest amplitude. (c) Mode n = 1, m = 3 has the largest amplitude.

Grahic Jump Location
Fig. 5

(a) Prediction of graphene layer thermal expansion coefficient. Dots are the result of our semi-analytic method; lines are experimental results, by Yoon et al. [33], Bao et al. [4], and Pan et al. [34]; and line is the theoretical work by Mounet and Marzari [48]. The inset shows α as a function of T over a broader temperature range. (b) Prediction of graphene layer thermal expansion coefficient under distinct shear loading (upper panel) and hydrostatic tension (lower panel); the unit in the legends is pN nm−1.

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